U.S. Pat. No. 4,372,173 relates to the selection of beam dimensions for avoiding the development of spurious modes of oscillation in both single beam and double beam vibratory transducers. For a transducer beam having a thickness "t", width "w", length "1", and distance "m" between the locations at which the beam ends are coupled together, the ratios "t/w" and "1/m" must be selected so that the desired characteristic resonant frequency "f" and multiples thereof for the transducer, do not equal one of the spurious mode resonant frequencies anywhere in the operating range, i.e. for any sensitivity values in the desired range of sensitivity of the force transducer. Sensitivity of a transducer is its ability to detect changes in an applied force F and can be defined as the fractional change in frequency with applied force. To achieve high sensitivity, the patent discloses "t" and "w" should be selected in the range such that 0.4&lt;t/w &lt;4 although certain values of t/w in this range including t=w are precluded. For minimum stress consistent with maximum sensitivity, however, t/w should be chosen to be as large as possible. This specifies values of t which are greater than w, but in practice, it is not practical to fabricate a beam transducer by photo-etching with values of t&gt;w.
While quartz, a prezoelectric material, is widely used in vibratory beam transducers because of demonstrated high mechanical "Q", dimensional stability, good elastic characteristics, and low temperature coefficient of expansion, its disadvantages pertaining to fragility, and difficulty to produce and handle in small dimensional sensors has led to the development and use of metal vibratable beams in resonator transducers. Such beams can be fabricated by an electro-etching process with great accuracy for very small dimensional transducers.
There are also many transducers which are used in applications where only tensile forces are axially applied to the vibrating beam and problems associated with "buckling" of the beam are not present. Considerably superior transducers can then be designed when design restrictions for avoiding buckling are eliminated.